Question: $g'(x)=\dfrac{3x}{4g(x)}$ Is $g(x)=\dfrac{\sqrt{3x^2}}{2}$ a solution to the above equation? Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No
In order to find whether $g(x)=\dfrac{\sqrt{3x^2}}{2}$ is a solution, we need to substitute it into the equation and see if we get equivalent expressions on each side of the equal sign. In addition to substituting for $g(x)$, we need to find the corresponding $g'(x)$ expression to substitute into the equation: $\begin{aligned} g'(x)&=\dfrac{d}{dx}\left[\dfrac{\sqrt{3x^2}}{2}\right] \\\\ &=\dfrac{3x}{2\sqrt{3x^2}} \end{aligned}$ Now we substitute ${g(x)=\dfrac{\sqrt{3x^2}}{2}}$ and ${g'(x)=\dfrac{3x}{2\sqrt{3x^2}}}$ into the equation: $\begin{aligned} {g'(x)}&=\dfrac{3x}{4{g(x)}} \\\\ {\dfrac{3x}{2\sqrt{3x^2}}}&\stackrel{?}{=}\dfrac{3x}{4\left({\dfrac{\sqrt{3x^2}}{2}}\right)} \\\\ \dfrac{3x}{2\sqrt{3x^2}}&\stackrel{\checkmark}{=}\dfrac{3x}{2\sqrt{3x^2}} \end{aligned}$ We obtained the same expression on each side. In conclusion, yes, $g(x)=\dfrac{\sqrt{3x^2}}{2}$ is a solution to the differential equation.